There are 20 students in a science class. All are seience majors - each majors in Chemistry, Physics, or Mathematics, including some who major in two or even three of those subjects. Of those 20 students:
10 major in Chemistry (and, possibly, in other subjects), 13 major in Physics (and. possibly, in other subjects),
7 major in both Chemistry and Physics (and, possibly, in Math),
6 major in both Chemistry and Mathematics (and, possibly, in Physics), 8 major in both Mathematics and Physics (and, possibly, in Chemistry),
5 major in all the three subjects.
(a) How many students major in Mathematies?
(b) If a random Mathematies major is selected, what is the probability that this student majors in exactly two subjects?

This problem involves using the inclusion-exclusion principle to count the number of students majoring in each subject, along with some set theory to answer the given questions. Let's solve the problem step by step.

### Step 1: Define the Sets
Let:
- \( C \) be the set of students majoring in **Chemistry**,
- \( P \) be the set of students majoring in **Physics**,
- \( M \) be the set of students majoring in **Mathematics**.

We are given the following information:
- \( |C| = 10 \) (students majoring in Chemistry),
- \( |P| = 13 \) (students majoring in Physics),
- \( |C \cap P| = 7 \) (students majoring in both Chemistry and Physics),
- \( |C \cap M| = 6 \) (students majoring in both Chemistry and Mathematics),
- \( |P \cap M| = 8 \) (students majoring in both Physics and Mathematics),
- \( |C \cap P \cap M| = 5 \) (students majoring in all three subjects),
- The total number of students is \( 20 \).

### Step 2: Calculate the Number of Students Majoring in Mathematics
We will use the inclusion-exclusion principle to find \( |M| \), the number of students majoring in Mathematics.

\[
|M| = |C \cap M| + |P \cap M| - |C \cap P \cap M| + \text{(students who only major in Mathematics)}
\]

We can rearrange this to:

\[
|M| = |C \cap M| + |P \cap M| - |C \cap P \cap M| + x
\]

Where \( x \) is the number of students who only major in Mathematics. 

First, let’s figure out the total number of students in all categories using the inclusion-exclusion principle:

\[
|C \cup P \cup M| = |C| + |P| + |M| - |C \cap P| - |C \cap M| - |P \cap M| + |C \cap P \cap M|
\]

Substitute the known values:

\[
20 = 10 + 13 + |M| - 7 - 6 - 8 + 5
\]

Simplifying this:

\[
20 = 22 + |M| - 16
\]
\[
20 = 6 + |M|
\]
\[
|M| = 14
\]

Thus, the number of students majoring in Mathematics is \( \boxed{14} \).

### Step 3: Probability of Majoring in Exactly Two Subjects
To find the probability that a randomly selected Mathematics major is majoring in exactly two subjects, we first need to determine how many Mathematics majors are in exactly two subjects.

From the problem, the students majoring in two subjects are:
- \( |C \cap M| - |C \cap P \cap M| = 6 - 5 = 1 \) (students majoring in Chemistry and Mathematics but not Physics),
- \( |P \cap M| - |C \cap P \cap M| = 8 - 5 = 3 \) (students majoring in Physics and Mathematics but not Chemistry).

So, the total number of Mathematics majors in exactly two subjects is:

\[
1 + 3 = 4
\]

Now, the probability that a randomly selected Mathematics major is in exactly two subjects is:

\[
\text{Probability} = \frac{\text{Number of Mathematics majors in exactly two subjects}}{\text{Total number of Mathematics majors}} = \frac{4}{14} = \frac{2}{7}
\]

Thus, the probability is \( \boxed{\frac{2}{7}} \).

---

Do you want more details or have any questions?

Here are five related questions:
1. How many students major in only one subject?
2. What is the probability that a randomly selected Physics major also majors in Chemistry?
3. How many students major in exactly three subjects?
4. How many students major only in Mathematics?
5. What is the total number of students who major in more than one subject?

**Tip**: The inclusion-exclusion principle is very useful when dealing with sets and overlapping categories.

There are 20 students in a science class. All are science majors - each majors in Chemistry, Physics, or Mathematics, including some who major in two or even three of those subjects. Of those 20 students: 10 major in Chemistry (and, possibly, in other subjects), 13 major in Physics (and, possibly, in other subjects), 7 major in both Chemistry and Physics (and, possibly, in Math), 6 major in both Chemistry and Mathematics (and, possibly, in Physics), 8 major in both Mathematics and Physics (and, possibly, in Chemistry), 5 major in all the three subjects. (a) How many students major in Mathematics? (b) If a random Mathematics major is selected, what is the probability that this student majors in exactly two subjects?

Solve a problem using the inclusion-exclusion principle to count students majoring in Chemistry, Physics, and Mathematics. Learn how to apply set theory and probability to find the number of Mathematics majors and the probability of selecting a student majoring in exactly two subjects. Suitable for Grades 10-12.

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